Determining the Terminal Point

Oct 13, 2025 by windailys.com 57 views

Introduction

In trigonometry, the sine function is a fundamental concept used to describe the relationship between the angles and side lengths of triangles. Given the values of $\sin t = 0.45$ and $\sin w = 0.89$ , we are tasked with determining a terminal point in Quadrant 1 and understanding the relationship between $t$ and $w$ . In this article, we will delve into the world of trigonometry and explore the properties of the sine function to determine the correct relationship between $t$ and $w$ .

The Sine Function

The sine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function is also related to the unit circle, where the sine of an angle is equal to the y-coordinate of the point on the unit circle corresponding to that angle.

Given Values

We are given that $\sin t = 0.45$ and $\sin w = 0.89$ . Both $t$ and $w$ are positive angles, and we are asked to determine a terminal point in Quadrant 1.

Determining the Terminal Point

To determine the terminal point in Quadrant 1, we need to find the values of $t$ and $w$ that correspond to the given sine values. We can use the inverse sine function to find the values of $t$ and $w$ .

import math

# Given values
sin_t = 0.45
sin_w = 0.89

# Find the values of t and w using the inverse sine function
t = math.asin(sin_t)
w = math.asin(sin_w)

# Convert the values to degrees
t_deg = math.degrees(t)
w_deg = math.degrees(w)

print(f"t = {t_deg} degrees")
print(f"w = {w_deg} degrees")

Understanding the Relationship

Now that we have found the values of $t$ and $w$ , we can determine the relationship between them. We can use the fact that the sine function is periodic to find the relationship between $t$ and $w$ .

# Find the relationship between t and w
relationship = f"{w_deg} = {t_deg} + 360k"
print(relationship)

Conclusion

In this article, we have determined a terminal point in Quadrant 1 and understood the relationship between $t$ and $w$ . We have used the inverse sine function to find the values of $t$ and $w$ and then used the periodic property of the sine function to determine the relationship between them. The relationship between $t$ and $w$ is given by $w = t + 360k$ , where $k$ is an integer.

Relationship Between Trigonometric Functions

The relationship between trigonometric functions is a fundamental concept in mathematics. In this article, we have explored the properties of the sine function and used it to determine the relationship between $t$ and $w$ . The relationship between trigonometric functions is essential in understanding many mathematical concepts, including calculus, differential equations, and number theory.

Key Takeaways

The sine function is a periodic function that oscillates between -1 and 1.
The sine function is related to the unit circle, where the sine of an angle is equal to the y-coordinate of the point on the unit circle corresponding to that angle.
The inverse sine function can be used to find the values of angles that correspond to given sine values.
The periodic property of the sine function can be used to determine the relationship between angles.

References

"Trigonometry" by Michael Corral
"Calculus" by Michael Spivak
"Differential Equations" by Lawrence Perko

Appendix

The following is a list of common trigonometric identities:

$\sin^2 x + \cos^2 x = 1$
$\tan x = \frac{\sin x}{\cos x}$
$\cot x = \frac{\cos x}{\sin x}$

Introduction

In our previous article, we explored the properties of the sine function and used it to determine the relationship between $t$ and $w$ . In this article, we will answer some frequently asked questions about trigonometry and the relationship between trigonometric functions.

Q&A

Q: What is the relationship between the sine and cosine functions?

A: The sine and cosine functions are related by the Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ . This identity shows that the sine and cosine functions are complementary and that their squares add up to 1.

Q: How do I find the value of an angle given its sine or cosine value?

A: To find the value of an angle given its sine or cosine value, you can use the inverse sine or inverse cosine function. For example, if you know that $\sin x = 0.5$ , you can use the inverse sine function to find the value of $x$ .

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both periodic functions that oscillate between -1 and 1. However, the sine function is related to the y-coordinate of the point on the unit circle, while the cosine function is related to the x-coordinate.

Q: How do I use the Pythagorean identity to find the value of an angle?

A: To use the Pythagorean identity to find the value of an angle, you can rearrange the identity to solve for the sine or cosine function. For example, if you know that $\sin x = 0.5$ and $\cos x = 0.866$ , you can use the Pythagorean identity to find the value of $x$ .

Q: What is the relationship between the tangent and cotangent functions?

A: The tangent and cotangent functions are related by the identity: $\tan x = \frac{\sin x}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$ . This identity shows that the tangent and cotangent functions are reciprocal functions.

Q: How do I use the tangent and cotangent functions to find the value of an angle?

A: To use the tangent and cotangent functions to find the value of an angle, you can use the inverse tangent or inverse cotangent function. For example, if you know that $\tan x = 1$ , you can use the inverse tangent function to find the value of $x$ .

Q: What is the difference between the sine and cosine functions in terms of their periodicity?

A: The sine and cosine functions are both periodic functions that oscillate between -1 and 1. However, the sine function has a period of $2\pi$ , while the cosine function has a period of $2\pi$ as well.

Q: How do I use the periodicity of the sine and cosine functions to find the value of an angle?

A: To use the periodicity of the sine and cosine functions to find the value of an angle, you can use the fact that the sine and cosine functions repeat every $2\pi$ radians. For example, if you know that $\sin x = 0.5$ , you can use the periodicity of the sine function to find the value of $x$ .

Conclusion

In this article, we have answered some frequently asked questions about trigonometry and the relationship between trigonometric functions. We have explored the properties of the sine and cosine functions, including their relationship, periodicity, and use in finding the value of an angle. We hope that this article has been helpful in understanding the relationship between trigonometric functions.

Key Takeaways

The sine and cosine functions are related by the Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ .
The sine and cosine functions are periodic functions that oscillate between -1 and 1.
The tangent and cotangent functions are related by the identity: $\tan x = \frac{\sin x}{\cos x}$ and $\cot x = \frac{\cos x}{\sin x}$ .
The periodicity of the sine and cosine functions is $2\pi$ radians.

References

"Trigonometry" by Michael Corral
"Calculus" by Michael Spivak
"Differential Equations" by Lawrence Perko

Appendix

The following is a list of common trigonometric identities:

$\sin^2 x + \cos^2 x = 1$
$\tan x = \frac{\sin x}{\cos x}$
$\cot x = \frac{\cos x}{\sin x}$

These identities are essential in understanding many mathematical concepts, including calculus, differential equations, and number theory.